What is the integral of (e^{3x}+1)/(e^{2x)} ?

The integral of (e^{3x}+1)/(e^{2x)} is e^x-1/2 e^{-2x}+C

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integral of (e^{3x}+1)/(e^{2x)}

\int \frac{e ^{3 x} + 1}{e ^{2 x}} d x

e^{x} - \frac{1}{2} e^{- 2 x} + C

Solution steps

\int \frac{e ^{3 x} + 1}{e ^{2 x}} d x

Apply Integration By Parts

: - \frac{1}{2} e^{- 2 x} (e^{3 x} + 1) - \int - \frac{3}{2} e^{x} d x

= - \frac{1}{2} e^{- 2 x} (e^{3 x} + 1) - \int - \frac{3}{2} e^{x} d x

\int - \frac{3}{2} e^{x} d x = - \frac{3}{2} e^{x}

= - \frac{1}{2} e^{- 2 x} (e^{3 x} + 1) - (- \frac{3}{2} e^{x})

Simplify

- \frac{1}{2} e^{- 2 x} (e^{3 x} + 1) - (- \frac{3}{2} e^{x}) : e^{x} - \frac{1}{2} e^{- 2 x}

= e^{x} - \frac{1}{2} e^{- 2 x}

Add a constant to the solution

= e^{x} - \frac{1}{2} e^{- 2 x} + C

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What is the integral of (e^{3x}+1)/(e^{2x)} ?
The integral of (e^{3x}+1)/(e^{2x)} is e^x-1/2 e^{-2x}+C